cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 18 results. Next

A134605 Composite numbers such that the square root of the sum of squares of their prime factors (with multiplicity) is an integer.

Original entry on oeis.org

16, 48, 81, 320, 351, 486, 512, 625, 1080, 1260, 1350, 1375, 1792, 1836, 2070, 2145, 2175, 2401, 2730, 2772, 3072, 3150, 3510, 4104, 4305, 4625, 4650, 4655, 4998, 5880, 6000, 6174, 6545, 7098, 7128, 7182, 7650, 7791, 7889, 7956, 9030, 9108, 9295, 9324
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Examples

			a(2)=48 since 48=2*2*2*2*3 and sqrt(4*2^2+3^2)=sqrt(25)=5.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376, A134600, A134602, A134608, A134611, A134616, A134618, A134620.

Programs

  • Mathematica
    srssQ[n_]:=IntegerQ[Sqrt[Total[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]]^2]]]; Select[Range[10000],CompositeQ[#]&&srssQ[#]&] (* Harvey P. Dale, Jan 22 2019 *)
  • PARI
    is(n)=my(f=factor(n)); issquare(sum(i=1,#f~,f[i,1]^2*f[i,2])) && !isprime(n) && n>1 \\ Charles R Greathouse IV, Apr 29 2015

A134608 Composite numbers such that the cube root of the sum of cubes of their prime factors is an integer.

Original entry on oeis.org

256, 588, 693, 3840, 6561, 14157, 17787, 141960, 178360, 313600, 337365, 350000, 387072, 390625, 407442, 432000, 466560, 531674, 535815, 541310, 664909, 697851, 1044582, 1262056, 1264640, 1299272, 1374327, 1547570, 1608575, 1660360
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Examples

			a(3)=693, since 693=3*3*7*11 and (2*3^3+7^3+11^3)^(1/3)=1728^(1/3)=12.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134611, A134616, A134618, A134620.

Programs

  • Mathematica
    criQ[n_]:=IntegerQ[Surd[Total[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]]^3],3]]; Select[Range[1670000],CompositeQ[#] && criQ[#]&] (* Harvey P. Dale, Sep 19 2021 *)
  • PARI
    lista(m) = {for (i=2, m, if (! isprime(i), f = factor(i); s = sum (j=1, length(f~), f[j,1]^3*f[j,2]); if (ispower(s, 3), print1(i, ", "));););} \\ Michel Marcus, Apr 14 2013

Extensions

Minor edits by Hieronymus Fischer, Apr 20 2013

A134621 Numbers such that the arithmetic mean of the 4th power of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

15, 28, 39, 48, 51, 65, 68, 76, 77, 85, 87, 93, 111, 119, 133, 141, 143, 148, 155, 161, 175, 187, 189, 209, 212, 215, 221, 225, 235, 244, 275, 287, 295, 301, 315, 316, 320, 323, 329, 355, 393, 403, 404, 411, 428, 437, 447, 451, 455, 481, 505, 508, 515, 517
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Examples

			a(3)=39, since 39=3*13 and (3^4+13^4)/2=14321 which is prime.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134613, A134617, A134619.

Extensions

Minor edits by Hieronymus Fischer, May 06 2013

A134619 Numbers such that the arithmetic mean of the cubes of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

20, 44, 188, 297, 336, 400, 425, 540, 575, 605, 704, 752, 764, 908, 912, 1025, 1053, 1124, 1172, 1183, 1365, 1380, 1412, 1420, 1452, 1475, 1484, 1519, 1604, 1625, 1809, 1844, 1856, 1936, 1953, 2107, 2192, 2205, 2255, 2320, 2325, 2348, 2368, 2372, 2468
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Examples

			a(10)=605, since 605=5*11*11 and (5^3+11^3+11^3)/3=929 which is prime.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134613, A134617, A134621.

Programs

  • Mathematica
    amcpfQ[n_]:=PrimeQ[Mean[Flatten[PadRight[{},#[[2]],#[[1]]]&/@FactorInteger[n]]^3]]; Select[ Range[ 2500],amcpfQ] (* Harvey P. Dale, Jun 06 2023 *)
  • PARI
    lista(m) = {for (i=2, m, f = factor(i); s = sum (j=1, length(f~), f[j,1]^3*f[j,2]); s /= bigomega(i); if (type(s) == "t_INT" && isprime(s), print1(i, ", ")););} \\ Michel Marcus, Apr 14 2013

Extensions

Minor edits by Hieronymus Fischer, May 06 2013

A134611 Nonprime numbers such that the root mean cube of their prime factors is an integer (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1512, 1681, 1849, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

The prime factors are taken with multiplicity.
All perfect prime powers (A025475) are included. First term not included in A025475 is a(30) = 1512 = A134613(2) = A134613(1).
Most terms have a last digit of 1 or 9 (i.e., 8326 out of 9000 terms). Mainly, this comes from the fact that all squares of primes are included. Since each prime > 10 has a last digit of 1, 3, 7 or 9, its square has a last digit of 1 or 9. In addition, m-th powers of primes have a last digit of 1, if m == 0 (mod 4), and have a last digit of 1 or 9 if m == 2 (mod 4), and have a 50% chance, roughly, for a last digit of 1 or 9, if m == 1 (mod 4) or m == 3 (mod 4). Since the number of terms <= N which are squares of primes is PrimePi(sqrt(N)) = A000720(sqrt(N)), it follows that the number of terms <= N which have a last digit of 1 or 9 is greater than PrimePi(sqrt(N)). This can be estimated as 2*N^(1/2)/log(N), approximately.

Examples

			a(6) = 25, since 25 = 5*5 and ((5^3+5^3)/2)^(1/3) = 5.
a(30) = 1512, since 1512 = 2*2*2*3*3*3*7 and ((3*2^3+3*3^3+7^3)/7)^(1/3) = 64^(1/3) = 4.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134617, A134619, A134621.

Programs

  • PARI
    lista(m) = {for (i=2, m, if (! isprime(i), f = factor(i); s = sum (j=1, length(f~), f[j,1]^3*f[j,2]); s /= bigomega(i); if (type(s) == "t_INT" && ispower(s, 3), print1(i, ", "));););}  \\ Michel Marcus, Apr 14 2013

Extensions

Edited by Hieronymus Fischer, May 30 2013

A134617 Numbers such that the arithmetic mean of the squares of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

15, 20, 21, 28, 35, 39, 44, 48, 51, 52, 55, 65, 69, 85, 91, 92, 95, 108, 112, 115, 116, 129, 135, 141, 145, 159, 164, 172, 188, 189, 205, 208, 209, 215, 221, 225, 235, 236, 245, 249, 259, 268, 272, 295, 297, 299, 305, 309, 315, 316, 320, 325, 329, 339, 341, 365
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Examples

			a(2)=20, since 20=2*2*5 and (2^2+2^2+5^2)/3=33/3=11.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134613, A134619, A134621.

Programs

  • Mathematica
    amspQ[n_]:=PrimeQ[Mean[Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[ n]]^2]]; Select[Range[400],amspQ] (* Harvey P. Dale, Jan 21 2017 *)

Extensions

Minor edits by the author, May 06 2013

A134616 Numbers such that the sum of squares of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

6, 10, 12, 14, 26, 34, 40, 45, 54, 56, 63, 74, 75, 80, 90, 94, 96, 99, 104, 105, 126, 134, 146, 147, 152, 153, 171, 176, 184, 194, 206, 207, 231, 232, 234, 250, 261, 273, 274, 296, 300, 306, 326, 328, 334, 342, 344, 345, 350, 357, 363, 369, 376, 384, 386, 387
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Examples

			a(2)=10, since 10=2*5 and 2^2+5^2=29 which is prime.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134612, A134619, A134621.

Programs

  • Mathematica
    f[{a_,b_}]:=Table[a,b];Select[Range[2,387],PrimeQ[ Total[Flatten[(f/@FactorInteger[#])^2]] ]&] (* James C. McMahon, Apr 09 2025 *)

Extensions

Minor edits by the author, May 06 2013

A134618 Numbers such that the sum of cubes of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

12, 28, 40, 45, 48, 52, 54, 56, 63, 75, 80, 96, 104, 108, 117, 136, 152, 153, 165, 175, 210, 224, 232, 245, 250, 261, 268, 300, 320, 325, 333, 344, 350, 363, 384, 387, 390, 399, 405, 416, 432, 462, 464, 468, 475, 477, 504, 507, 531, 536, 539, 561, 570, 584
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Examples

			a(2) = 28, since 28 = 2*2*7 and 2^3 + 2^3 + 7^3 = 359 which is prime.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134612, A134616, A134620.

Programs

  • Mathematica
    Select[Range[600],PrimeQ[Total[Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[#]]^3]]&] (* Harvey P. Dale, Feb 01 2013 *)
  • Python
    from sympy import factorint, isprime
def ok(n): return isprime(sum(p**3 for p in factorint(n, multiple=True)))
print([k for k in range(585) if ok(k)]) # Michael S. Branicky, Dec 28 2021

Formula

{k: A224787(k) in A000040}. - R. J. Mathar, Mar 25 2025

Extensions

Example clarified by Harvey P. Dale, Feb 01 2013
Minor edits by Hieronymus Fischer, May 06 2013

A134620 Numbers such that the sum of 4th power of their prime factors is a prime.

Original entry on oeis.org

6, 10, 12, 14, 22, 34, 38, 40, 45, 46, 74, 82, 117, 118, 122, 126, 142, 152, 158, 171, 194, 231, 262, 278, 296, 345, 358, 363, 376, 384, 387, 429, 432, 446, 454, 458, 482, 486, 490, 500, 507, 522, 536, 550, 566, 584, 626, 627, 634, 639, 663, 675, 686, 704, 705
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

Prime factors must be taken with multiplicity. - Harvey P. Dale, May 23 2012
The calculation of higher terms is time-consuming, since for any number of the form 2*p with a prime number p > 10^5 the primality test have to be accomplished for a number > 10^20. - Hieronymus Fischer, May 21 2013

Examples

			a(2) = 10, since 10 = 2*5 and 2^4+5^4 = 641 which is prime.
a(9) = 45, since 45 = 3*3*5 and 3^4+3^4+5^4 = 787 which is prime.
a(9883) = 333314, since 333314 = 3*166657 and 2^4+166657^4 = 771425941499397811217 which is prime.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134612, A134616, A134618, A134621.

Programs

  • Mathematica
    Select[Range[1000],PrimeQ[Total[Flatten[Table[First[#],{Last[#]}]&/@ FactorInteger[#]]^4]]&] (* Harvey P. Dale, May 23 2012 *)

A134612 Nonprime numbers such that the root mean cube of their prime factors is a prime (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1681, 1849, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

The prime factors are taken with multiplicity.
All perfect prime powers (A025475) with power > 1 are included. First term not included in A025475 is a(211) = 707265 = A134614(5) = A134615(1).
Originally, the first term was 1. This was wrong, since the cube mean of the prime factors of 1 is zero, by definition of the empty sum.

Examples

			a(5) = 25, since 25 = 5*5 and ((5^3+5^3)/2)^(1/3) = 5.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134614, A134617, A134619, A134621.

Programs

  • PARI
    lista(m) = {for (i=2, m, if (! isprime(i), f = factor(i); s = sum (j=1, length(f~), f[j,1]^3*f[j,2]); s /= bigomega(i); if (type(s) == "t_INT" && ispower(s, 3, &p) && isprime(p), print1(i, ", "));););}  \\ Michel Marcus, Apr 14 2013

Extensions

Edited by Hieronymus Fischer, May 30 2013
Showing 1-10 of 18 results. Next

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